Version 10 (modified by 13 years ago) ( diff ) | ,
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Self Gravity
(This is a skeleton while a new version of this page is under construction.)
Physics
Implementation
Boundary Conditions
Present Considerations
Some notes on periodic BC's and particles
With periodic boundary conditions, we would like the total potential to be unchanged under the conversion of gas to particle. This requires adjusting the particle potential so that it has the same constant of integration as well as being periodic. When hypre solves for the potential, the constant of integration is a free parameter and I believe hypre adjusts it so that
. To calculate the potential of a point charge, we need the Greens function corresponding towhere D is the dimenesion of the problem. For 3D,
and for 2D,
and for 1D,
Now with periodic BC's, the solution to the gas potential is given by
, so the solution to the potential would actually be given by where is the mass of the particle. Note this is not the same as using mirror versions of the particle… An easier way to solve this however, is to use fourier transforms.
with . Now is just the fourier transform of a delta function times the mass, which is , so and
This can be discretized as